Through elementary operations, prove that the rank of above matrice is 2

To prove that the rank of the matrix is 2 using elementary row operations, we will convert the matrix into its row echelon form (REF) and then count the number of non-zero rows. The number of non-zero rows in REF gives the rank of the matrix.

Starting with the matrix:

\[
\begin{bmatrix}
2 & 4 & 8 & 6 \\
1 & 2 & 1 & 0 \\
3 & 6 & 6 & 3
\end{bmatrix}
\]

Let's apply the elementary row operations step by step.from sympy import Matrix

# Convert the matrix to a sympy Matrix to perform elementary row operations
matrix_sympy = Matrix([[2, 4, 8, 6],
                       [1, 2, 1, 0],
                       [3, 6, 6, 3]])

# Perform row reduction to get the row echelon form (REF)
ref_matrix = matrix_sympy.echelon_form()
ref_matrixAfter performing elementary row operations, the matrix in row echelon form (REF) is:

\[
\begin{bmatrix}
2 & 4 &  8 &  6 \\
0 & 0 & -6 & -6 \\
0 & 0 &  0 &  0
\end{bmatrix}
\]

In this form, we observe that there are **2 non-zero rows**. Therefore, the rank of the matrix is \( 2 \), as previously determined.

Would you like to explore more about the steps or the theory behind the row reduction process? Here are some related questions:

1. What elementary row operations were used to reduce the matrix?
2. Why do zero rows not contribute to the rank of a matrix?
3. How can the rank of a matrix help determine whether a system of equations has a unique solution?
4. What is the difference between rank and nullity in a matrix?
5. How does the echelon form relate to solving linear systems?

**Tip:** When reducing a matrix to row echelon form, pivot elements (the leading non-zero entries in each row) play a key role in determining the rank.

Learn how to prove the rank of a matrix is 2 using elementary row operations and row echelon form (REF). This problem involves matrix operations and is suitable for advanced high school students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation